I'm going to ask this question first in classical setting and then sketch its natural geometric setting.
Let $\Gamma$ be a subgroup of $\operatorname{PSL}_2(\mathbb Z)$ (the question is mostly interesting in the case where $\Gamma$ is a thin subgroup, i.e. non-elementary and of infinite index). Let $o$ be any point in the Poincaré half-plane $\mathbb H^2$. Define the Poincaré series $$ p_o(s) = \sum_{\gamma \in \Gamma} e^{-s \cdot d(o, \gamma o)}. $$ This always converges for $s > 1$ and the smallest $s$ for which it converges is called the critical exponent of $\Gamma$.
On the other hand let $\Gamma_\infty$ be the subgroup $$ \Gamma_\infty = \Gamma \cap \left\{ \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} :\: x \in \mathbb Z \right\} $$ and let $$ q_o(s) = \sum_{\gamma \in \Gamma/\Gamma_\infty} \operatorname{Im}(\gamma o)^s $$ which also converges for $s > 1$.
I am fairly certain that the infimal abscissa of convergence for the series $q_o(s)$ is the critical exponent of $\Gamma$, and that the behaviour at the critical exponent is the same. At least for the equality of abscissae of convergence i think this can be proved directly, by elementary estimates. I am also wondering whether one could recover $q_o$ from (variants of) $p_o$ through an asymptotic process. Does anybody know if this was written up somewhere? (The few places i looked at seem to consider both kind of series independently and not attempt to relate them.)
Of course, in the question above one can replace :
$\mathbb H^2$ with a proper Gromov-hyperbolic space $X$;
$\Gamma$ with a discrete subgroup of isometries of $X$;
$z\mapsto \operatorname{Im}(z)$ with (the exponential of) a Busemann function at a parabolic fixed point of $\Gamma$.
I would think that everything works fine if the stabilisers in $\Gamma$ of points at infinity of $X$ are amenable (i don't know if that is always the case in these circumstances).
